As the counterpart of conformal geometry, modular geometry on noncommutative manifolds explores the basic notions such as metric and curvature in Riemannian geometry (e.g. noncommutative tori) in a purely spectral framework. It was initiated by Connes-Tretkoff’s Gauss-Bonnet theorem on noncommutative two tori. Another milestone is the construction of modular Gaussian curvature due to Connes-Moscovici, which is derived from variation of the second heat coefficient of some Laplacian type operator. In this talk, I would like to report a few observations on the general structures of those heat coefficients. The word “modular” refers to the new ingredient of the coefficients, arising from the interaction between modular automorphisms associated to the volume state and the underlying smooth structure of the noncommutative manifolds. More precisely, one has to upgrade coefficients of local differential expressions from scalars to so-called rearrangement operators that fix various issues caused by the noncommutativity between metric coordinates and their derivatives. Like the notion of genus to a characteristic class, the spectral functions behind the rearrangement operators turn out to be intriguing. That is where hypergeometric functions come into play. The main result is the explicit formula of the second heat coefficient of a more general Laplacian type operator (beyond conformal perturbations studied in the literature). The talk is based on my recent preprint arxiv:2004.05714.

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